Preheat oven to 375 degrees. Stir together butter, brown sugar, oats, cinnamon, ginger, and salt in a small bowl. Set aside.
Scoop out the core of each apple with a melon baller, being careful not to go through the bottom; discard cores. Fill each apple with oat mixture, dividing evenly. Transfer apples to a 9-inch square non-reactive baking dish. Add 1 cup water to the dish to cover bottoms of apples by about 1/2 inch. Cover dish with aluminum foil. Bake until apples are soft but not mushy, 40 to 45 minutes. Serve warm.